Which of the following best describes the ultimate gain in the Ziegler-Nichols method?

• A. The gain at which the closed-loop system becomes stable with no oscillations.
• B. The gain that minimizes energy.
• C. The gain at which the closed-loop system has zero steady-state error.
• D. The gain at which the closed-loop system becomes marginally stable with sustained oscillations.

Answer: (D)

In this tuning method, you raise the proportional gain until the loop output exhibits continuous, constant-amplitude oscillations. that gain is called the ultimate gain. At that point the system sits right at the edge of stability—marginally stable. If you push the gain a bit higher, the oscillations grow and the system becomes unstable; if you drop the gain, the oscillations die out and the system is stable. This is why the ultimate gain is defined as the gain at which the closed-loop is marginally stable with sustained oscillations. The other ideas aren’t what this gain represents: stability with no oscillations occurs below this gain, energy minimization isn’t the criterion here, and zero steady-state error isn’t guaranteed by reaching the ultimate gain (that typically requires integral action).